"Du bout des lèvres" signifie dire quelque chose sans y croire vraiment. Il m´a dit qu´il viendrait mais il me l´a dit sur le bout des lèvres. (N´y compte pas trop.)

"monter aux lèvres" is a bit like "brûler les lèvres". It means she was thinking about saying something (bad or witty) and was struggling to refrain saying them. If you translate litteraly your expression, it means the words was raising to her lips. I'm not sure what would be the common expression in English, but I suggest something like "sprung to her mind".

I would say it was "on the tip of her tongue" You can say this if you feel you have to hold back in case you cause offence, but also if you have forgotten a word or name etc which is very familiar to you but you can't recall it on the spur of the moment. People say things like "I forget . . . . It's on the tip of my tongue, but I just can't remember it. I DO know it"

In English, I would suggest "came easily to mind" or "rolled off her tongue" ??